Two-Time-Scale Regime-Switching Stochastic Kolgomorov Systems With Wideband Noises

Abstract

In our recent work, in lieu of using white noise, we examined Kolmogorov systems driven by wideband noise. Such systems naturally arise in statistical physics, biological and ecological systems, and many related fields. One of the motivations of our study is to treat more realistic models than the usually assumed stochastic differential equation models. The rationale is that a Brownian motion is an idealization used in a wide range of models, whereas wideband noise processes are much easier to be realized in the actual applications. This paper further investigates the case that in addition to the wideband noise process, there is a singularly perturbed Markov chain. The added Markov chain is used to model discrete events. Although it is a more realistic formulation, because of the non-Markovian formulation due to the wideband noise and the singularly perturbed Markov chain, the analysis is more difficult. Using weak convergence methods, we obtain a limit result. Then we provide several examples for the utility of our findings.